Is there a contradiction between the continuity equation and Poiseuilles Law?

Question

The continuity equation states that flow rate should be conserved in different areas of a pipe:
$$Q = v_1 A_1 = v_2 A_2 = vpi r^2$$
We can see from this equation that velocity and pipe radius are inversely proportional. If radius is doubled, velocity of flow is quartered.
Another way I was taught to describe flow rate is through Poiseuilles Law:
$$Q = frac{pi r^4Delta P}{8eta L}$$
So if I were to plug in the continuity equations definition of flow rate into Poiseuilles Law:
$$vA = vpi r^2= frac{pi r^4Delta P}{8eta L} $$
Therefore:
$$v =  frac{ r^2Delta P}{8eta L}$$
Now in this case, the velocity is proportional to the radius of the pipe. If the radius is doubled, then velocity is qaudrupled.
What am I misunderstanding here? I would prefer a conceptual explanation because I feel that these equations are probably used with different assumptions/in different contexts.

Shrey · Answer

When you write down $Q = vA$, it's implicit that the velocity profile is uniform over the cross-section A (and purely perpendicular to it).
In general,
$$Q = int mathbf{v} cdot mathrm{d}mathbf{A}$$
This no longer implies that $v propto frac{1}{R^2}$.
If we assume $mathbf{v}$ has purely radial dependence and is aligned with $mathrm{d}mathbf{A}$ as in Poiseuille flow, then we have:
$$Q = 2pi int_0^R v(r) r mathrm{d}r$$

Chet Miller · Answer

For a fixed volumetric throughput rate Q, $Delta P$ decreases as $r^4$, so v decreases as $r^2$, exactly what you would expect from the continuity equation.

Is there a contradiction between the continuity equation and Poiseuilles Law?

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